Find dy/dx for x sec y = 4y - y tan 2x.

Step-by-step solution

1. 1

   Treat $y$ as a function of $x$ throughout. The equation cannot be solved for $y$ in closed form, so we differentiate both sides with respect to $x$ and let every $y$ carry a hidden $\dfrac{dy}{dx}$ from the chain rule. Both sides contain products of an $x$-factor with a $y$-factor, so the product rule appears twice.

2. 2

   Differentiate the left side. With $u=x$ and $v=\sec y$, and using $\dfrac{d}{dx}\sec y=\sec y\tan y\,\dfrac{dy}{dx}$: $\frac{d}{dx}\left(x\sec y\right)=\sec y + x\sec y\tan y\,\frac{dy}{dx}.$

3. 3

   Differentiate the right side, keeping the inner factor of $2$. The term $4y$ gives $4\dfrac{dy}{dx}$. For $y\tan 2x$ the product rule plus the chain rule (note $\dfrac{d}{dx}\tan 2x = 2\sec^{2}2x$, not $\sec^{2}2x$) gives $\frac{d}{dx}\left(y\tan 2x\right)=\frac{dy}{dx}\tan 2x + 2y\sec^{2}2x,$ so the whole right side differentiates to $4\frac{dy}{dx}-\frac{dy}{dx}\tan 2x-2y\sec^{2}2x.$ Dropping that factor $2$ is the single most common error in this problem.

4. 4

   Collect the $\dfrac{dy}{dx}$ terms on one side. Equating the two derivatives, $\sec y + x\sec y\tan y\,\frac{dy}{dx}=4\frac{dy}{dx}-\frac{dy}{dx}\tan 2x-2y\sec^{2}2x,$ and moving all $\dfrac{dy}{dx}$ terms right and everything else left: $\sec y + 2y\sec^{2}2x=\frac{dy}{dx}\left(4-\tan 2x-x\sec y\tan y\right).$

5. 5

   Divide to isolate the derivative. $\frac{dy}{dx}=\frac{\sec y + 2y\sec^{2}2x}{4-\tan 2x-x\sec y\tan y}.$ The result legitimately depends on both $x$ and $y$, which is normal for an implicit curve: to get a number you need a point that actually satisfies the original equation.

6. 6

   Check against the implicit function theorem. Writing $F(x,y)=x\sec y-4y+y\tan 2x$, we have $F_{x}=\sec y+2y\sec^{2}2x$ and $F_{y}=x\sec y\tan y-4+\tan 2x$, and $\dfrac{dy}{dx}=-\dfrac{F_{x}}{F_{y}}$ reproduces the same expression. A numerical test confirms it: at $x=0.3$ the curve passes through $y=0.09084883$, and a centred difference on the implicitly solved $y(x)$ gives $0.38647161$, while the formula gives $0.38647161$.